Families of bitangent planes of space curves and minimal non-fibration families

TL;DR

This paper classifies families of bitangent planes of cone curves, relates them to minimal non-fibration families on certain algebraic surfaces, and provides algorithms for their computation and analysis.

Contribution

It introduces a classification of bitangent plane families for cone curves and connects them to minimal non-fibration families on weak Del Pezzo surfaces, with computational methods included.

Findings

Classification of bitangent families for cone curves.

Algorithms for computing bitangent families and their genera.

Identification of minimal non-fibration families on Del Pezzo surfaces.

Abstract

We define a cone curve to be a reduced sextic space curve which lies on a quadric cone and does not go through the vertex. We classify families of bitangent planes of cone curves. The methods we apply can be used for any space curve with ADE singularities, though in this paper we concentrate on cone curves. An embedded complex projective surface which is adjoint to a degree one weak Del Pezzo surface contains families of minimal degree rational curves, which cannot be defined by the fibers of a map. Such families are called minimal non-fibration families. Families of bitangent planes of cone curves correspond to minimal non-fibration families. The main motivation of this paper is to classify minimal non-fibration families. We present algorithms wich compute all bitangent families of a given cone curve and their geometric genera. We consider cone curves to be equivalent if they have the same singularity configuration. For each equivalence class of cone curves we determine the possible number of bitangent families and the number of rational bitangent families. Finally we compute an example of a minimal non-fibration family on an embedded weak degree one Del Pezzo surface.